216 ME. G. H. LIVENS ON THE 



necessarily imply that the integrand involved is a true Lagrangian function for the 

 system. 



6. Now let us apply these principles to our electromagnetic problem. The conditions 

 in the field surrounding a number of bodies are specified in the usual way by the 

 magnetic induction vector B and the electric force vector E, and the part of the 

 Lagrangian function associated with this field may be taken to be 



the first term denoting the magnetic or kinetic energy and the second the electric or 

 potential, and the integral is extended over the whole of space. In addition to these 

 energies there will tie the energies of the material bodies in the field which will consist 

 in part of the kinetic energies of their organised motions, in part of their potential 

 energy relative to one another or to any extraneous fields of non-electric nature, and 

 in part finally of internal energy of elastic or motional type in the media. The part 

 of the Lagrangian function corresponding to these energies can, in the most general 

 case, be denoted by 



where ,L is the Lagrangian function of the organised motions of the media, reckoned 

 per unit volume at each place and assumed to be a function only of the position 

 co-ordinates and velocities, and W, is the internal energy of all types reckoned as 

 potential energy per unit volume : this latter term will be a function of the electric 

 and magnetic polarisations in the media, but will be assumed not to depend to any 

 appreciable extent on the rates of variation of these conditions, and in so far as some 

 of the internal energy is essentially of kinetic type, it will be in reality a sort of 

 modified Lagrangian function with the energy corresponding to the motional terms 

 converted to potential energy in the usual way. The function L may also be taken to 

 include a part arising from the assumed inertia of any free electrons that may be 

 present. 



The motion of the system can now be expressed in the form 



dv 



and we could conduct the variation directly were it not for the fact that our functions 

 are not all expressed explicitly in terms of the independent co-ordinates of the 

 systems, which are in reality the position co-ordinates of the elements of matter and 

 electricity. As indicated above we can however avoid the use of any such explicit 

 interpretation by the use of undetermined multipliers. In this way the variations of 

 E and B can be temporarily rendered independent of each other and of the 

 actual co-ordinates of the material and electrical elements. 



